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Theorem List for Metamath Proof Explorer - 16101-16200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremasclpropd 16101* If two structures have the same components (properties), one is an associative algebra iff the other one is. The last hypotheses on  1r can be discharged either by letting  W  =  _V (if strong equality is known on  .s) or assuming  K is a ring. (Contributed by Mario Carneiro, 5-Jul-2015.)
 |-  F  =  (Scalar `  K )   &    |-  G  =  (Scalar `  L )   &    |-  ( ph  ->  P  =  ( Base `  F )
 )   &    |-  ( ph  ->  P  =  ( Base `  G )
 )   &    |-  ( ( ph  /\  ( x  e.  P  /\  y  e.  W )
 )  ->  ( x ( .s `  K ) y )  =  ( x ( .s `  L ) y ) )   &    |-  ( ph  ->  ( 1r `  K )  =  ( 1r `  L ) )   &    |-  ( ph  ->  ( 1r `  K )  e.  W )   =>    |-  ( ph  ->  (algSc `  K )  =  (algSc `  L ) )
 
Theoremaspval2 16102 The algebraic closure is the ring closure when the generating set is expanded to include all scalars. EDITORIAL : In light of this, is AlgSpan independently needed? (Contributed by Stefan O'Rear, 9-Mar-2015.)
 |-  A  =  (AlgSpan `  W )   &    |-  C  =  (algSc `  W )   &    |-  R  =  (mrCls `  (SubRing `  W )
 )   &    |-  V  =  ( Base `  W )   =>    |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  =  ( R `  ( ran  C  u.  S ) ) )
 
10.10  Abstract multivariate polynomials
 
10.10.1  Definition and basic properties
 
Syntaxcmps 16103 Multivariate power series.
 class mPwSer
 
Syntaxcmvr 16104 Multivariate power series variables.
 class mVar
 
Syntaxcmpl 16105 Multivariate polynomials.
 class mPoly
 
Syntaxces 16106 Evaluation in a superring.
 class evalSub
 
Syntaxcevl 16107 Evaluation of a multivariate polynomial.
 class eval
 
Syntaxcmhp 16108 Multivariate polynomials.
 class mHomP
 
Syntaxcpsd 16109 Power series partial derivative function.
 class mPSDer
 
Syntaxcltb 16110 Ordering on terms of a multivariate polynomial.
 class  <bag
 
Syntaxcopws 16111 Ordered set of power series.
 class ordPwSer
 
Syntaxcslv 16112 Select a subset of variables in a multivariate polynomial.
 class selectVars
 
Syntaxcai 16113 Algebraically independent.
 class AlgInd
 
Definitiondf-psr 16114* Define the algebra of power series over the index set  i and with coefficients from the ring  r. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- mPwSer  =  ( i  e.  _V ,  r  e.  _V  |->  [_
 { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin } 
 /  d ]_ [_ (
 ( Base `  r )  ^m  d )  /  b ]_ ( { <. ( Base ` 
 ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  r
 )  |`  ( b  X.  b ) ) >. , 
 <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r  gsumg  ( x  e.  { y  e.  d  |  y  o R  <_  k }  |->  ( ( f `  x ) ( .r
 `  r ) ( g `  ( k  o F  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  r >. , 
 <. ( .s `  ndx ) ,  ( x  e.  ( Base `  r ) ,  f  e.  b  |->  ( ( d  X.  { x } )  o F ( .r `  r ) f ) ) >. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( d  X.  {
 ( TopOpen `  r ) } ) ) >. } ) )
 
Definitiondf-mvr 16115* Define the generating elements of the power series algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |- mVar  =  ( i  e.  _V ,  r  e.  _V  |->  ( x  e.  i  |->  ( f  e.  { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  |->  if (
 f  =  ( y  e.  i  |->  if (
 y  =  x , 
 1 ,  0 ) ) ,  ( 1r
 `  r ) ,  ( 0g `  r
 ) ) ) ) )
 
Definitiondf-mpl 16116* Define the subalgebra of the power series algebra generated by the variables; this is the polynomial algebra (the set of power series with finite degree). (Contributed by Mario Carneiro, 7-Jan-2015.)
 |- mPoly  =  ( i  e.  _V ,  r  e.  _V  |->  [_ ( i mPwSer  r ) 
 /  w ]_ ( ws  { f  e.  ( Base `  w )  |  ( `' f " ( _V  \  { ( 0g `  r ) } )
 )  e.  Fin }
 ) )
 
Definitiondf-evls 16117* Define the evaluation map for the polynomial algebra. The function  ( (
I evalSub  S ) `  R
) : V --> ( S  ^m  ( S  ^m  I ) ) makes sense when  I is an index set,  S is a ring,  R is a subring of  S, and where  V is the set of polynomials in  ( I mPoly  R
). This function maps an element of the formal polynomial algebra (with coefficients in  R) to a function from assignments  I --> S of the variables to elements of  S formed by evaluating the polynomial with the given assignments. (Contributed by Stefan O'Rear, 11-Mar-2015.)
 |- evalSub  =  ( i  e.  _V ,  s  e.  CRing  |->  [_ ( Base `  s )  /  b ]_ ( r  e.  (SubRing `  s )  |-> 
 [_ ( i mPoly  (
 ss  r ) )  /  w ]_ ( iota_ f  e.  ( w RingHom  ( s  ^s  ( b  ^m  i ) ) ) ( ( f  o.  (algSc `  w ) )  =  ( x  e.  r  |->  ( ( b  ^m  i )  X.  { x } ) )  /\  ( f  o.  (
 i mVar  ( ss  r ) ) )  =  ( x  e.  i  |->  ( g  e.  ( b 
 ^m  i )  |->  ( g `  x ) ) ) ) ) ) )
 
Definitiondf-evl 16118* A simplication of evalSub when the evaluation ring is the same as the coefficient ring. (Contributed by Stefan O'Rear, 19-Mar-2015.)
 |- eval  =  ( i  e.  _V ,  r  e.  _V  |->  ( ( i evalSub  r
 ) `  ( Base `  r ) ) )
 
Definitiondf-mhp 16119* Define the subspaces of order-  n homogeneous polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- mHomP  =  ( i  e.  _V ,  r  e.  _V  |->  ( n  e.  NN0  |->  { f  e.  ( Base `  ( i mPoly  r ) )  |  ( `' f " ( _V  \  { ( 0g `  r ) } )
 )  C_  { g  e.  { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  |  sum_ j  e.  NN0  ( g `  j
 )  =  n } } ) )
 
Definitiondf-psd 16120* Define the differentiation operation on multivariate polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- mPSDer  =  ( i  e.  _V ,  r  e.  _V  |->  ( x  e.  i  |->  ( f  e.  ( Base `  ( i mPwSer  r
 ) )  |->  ( k  e.  { h  e.  ( NN0  ^m  i
 )  |  ( `' h " NN )  e.  Fin }  |->  ( ( ( k `  x )  +  1 )
 (.g `  r ) ( f `  ( k  o F  +  (
 y  e.  i  |->  if ( y  =  x ,  1 ,  0 ) ) ) ) ) ) ) ) )
 
Definitiondf-ltbag 16121* Define a well-order on the set of all finite bags from the index set  i given a wellordering  r of  i. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |- 
 <bag 
 =  ( r  e. 
 _V ,  i  e. 
 _V  |->  { <. x ,  y >.  |  ( { x ,  y }  C_  { h  e.  ( NN0  ^m  i
 )  |  ( `' h " NN )  e.  Fin }  /\  E. z  e.  i  (
 ( x `  z
 )  <  ( y `  z )  /\  A. w  e.  i  (
 z r w  ->  ( x `  w )  =  ( y `  w ) ) ) ) } )
 
Definitiondf-opsr 16122* Define a total order on the set of all power series in  s from the index set  i given a wellordering  r of  i and a totally ordered base ring  s. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |- ordPwSer  =  ( i  e.  _V ,  s  e.  _V  |->  ( r  e.  ~P ( i  X.  i
 )  |->  [_ ( i mPwSer  s
 )  /  p ]_ ( p sSet  <. ( le `  ndx ) ,  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  p )  /\  ( [. { h  e.  ( NN0  ^m  i
 )  |  ( `' h " NN )  e.  Fin }  /  d ]. E. z  e.  d  ( ( x `  z ) ( lt `  s ) ( y `
  z )  /\  A. w  e.  d  ( w ( r  <bag  i ) z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) } >. ) ) )
 
Definitiondf-selv 16123* Define the "variable selection" function. The function  ( (
I selectVars  R ) `  J
) maps elements of  ( I mPoly  R ) bijectively onto  ( J mPoly  ( ( I  \  J ) mPoly 
R ) ) in the natural way, for example if  I  =  { x ,  y } and  J  =  { y } it would map  1  +  x  +  y  +  x
y  e.  ( { x ,  y } mPoly 
ZZ ) to  ( 1  +  x )  +  ( 1  +  x ) y  e.  ( { y } mPoly  ( {
x } mPoly  ZZ )
). This, for example, allows one to treat a multivariate polynomial as a univariate polynomial with coefficients in a polynomial ring with one less variable. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- selectVars  =  ( i  e.  _V ,  r  e.  _V  |->  ( j  e.  ~P i  |->  ( f  e.  ( i mPoly  r ) 
 |->  [_ ( ( i 
 \  j ) mPoly  r
 )  /  s ]_ [_ ( x  e.  (Scalar `  s )  |->  ( x ( .s `  s
 ) ( 1r `  s ) ) ) 
 /  c ]_ (
 ( ( ( i evalSub  s ) `  (
 c  "s  r ) ) `  ( c  o.  f
 ) ) `  ( x  e.  i  |->  if ( x  e.  j ,  ( ( j mVar  (
 ( i  \  j
 ) mPoly  r ) ) `  x ) ,  (
 c  o.  ( ( ( i  \  j
 ) mVar  r ) `  x ) ) ) ) ) ) ) )
 
Definitiondf-algind 16124* Define the predicate "the set  v is algebraically independent in the algebra  w". A collection of vectors is algebraically independent if no nontrivial polynomial with elements from the subset evaluates to zero. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- AlgInd  =  ( w  e.  _V ,  k  e.  ~P ( Base `  w )  |->  { v  e.  ~P ( Base `  w )  |  Fun  `' ( f  e.  ( Base `  (
 v mPoly  ( ws  k ) ) ) 
 |->  ( ( ( ( v evalSub  w ) `  k
 ) `  f ) `  (  _I  |`  v ) ) ) } )
 
Theoremreldmpsr 16125 The multivariate power series constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- 
 Rel  dom mPwSer
 
Theorempsrval 16126* Value of the multivariate power series structure. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  K  =  (
 Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  O  =  ( TopOpen `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  B  =  ( K  ^m  D ) )   &    |-  .+b  =  (  o F  .+  |`  ( B  X.  B ) )   &    |-  .X. 
 =  ( f  e.  B ,  g  e.  B  |->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  { y  e.  D  |  y  o R  <_  k }  |->  ( ( f `  x )  .x.  ( g `
  ( k  o F  -  x ) ) ) ) ) ) )   &    |-  .xb  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  o F  .x.  f ) )   &    |-  ( ph  ->  J  =  (
 Xt_ `  ( D  X.  { O } )
 ) )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  X )   =>    |-  ( ph  ->  S  =  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+b  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s
 `  ndx ) ,  .xb  >. ,  <. (TopSet `  ndx ) ,  J >. } ) )
 
Theorempsrvalstr 16127 The multivariate power series structure is a function. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s
 `  ndx ) ,  .x.  >. ,  <. (TopSet `  ndx ) ,  J >. } ) Struct  <. 1 ,  9 >.
 
Theorempsrbag 16128* Elementhood in the set of finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( I  e.  V  ->  ( F  e.  D  <->  ( F : I --> NN0  /\  ( `' F " NN )  e.  Fin ) ) )
 
Theorempsrbagf 16129* A finite bag is a function. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( ( I  e.  V  /\  F  e.  D )  ->  F : I
 --> NN0 )
 
Theorempsrbaglesupp 16130* The support of a dominated bag is smaller than the dominating bag. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F ) ) 
 ->  ( `' G " NN )  C_  ( `' F " NN )
 )
 
Theorempsrbaglecl 16131* The set of finite bags is downward-closed. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F ) ) 
 ->  G  e.  D )
 
Theorempsrbagaddcl 16132* The sum of two finite bags is a finite bag. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( F  o F  +  G )  e.  D )
 
Theorempsrbagcon 16133* The analogue of the statement " 0  <_  G  <_  F implies  0  <_  F  -  G  <_  F " for finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F ) ) 
 ->  ( ( F  o F  -  G )  e.  D  /\  ( F  o F  -  G )  o R  <_  F ) )
 
Theorempsrbaglefi 16134* There are finitely many bags dominated by a given bag. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 25-Jan-2015.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( ( I  e.  V  /\  F  e.  D )  ->  { y  e.  D  |  y  o R  <_  F }  e.  Fin )
 
Theorempsrbagconcl 16135* The complement of a bag is a bag. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  S  =  {
 y  e.  D  |  y  o R  <_  F }   =>    |-  ( ( I  e.  V  /\  F  e.  D  /\  X  e.  S )  ->  ( F  o F  -  X )  e.  S )
 
Theorempsrbagconf1o 16136* Bag complementation is a bijection on the set of bags dominated by a given bag  F. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  S  =  {
 y  e.  D  |  y  o R  <_  F }   =>    |-  ( ( I  e.  V  /\  F  e.  D )  ->  ( x  e.  S  |->  ( F  o F  -  x ) ) : S -1-1-onto-> S )
 
Theoremgsumbagdiaglem 16137* Lemma for gsumbagdiag 16138. (Contributed by Mario Carneiro, 5-Jan-2015.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  S  =  {
 y  e.  D  |  y  o R  <_  F }   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F  e.  D )   =>    |-  ( ( ph  /\  ( X  e.  S  /\  Y  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  X ) }
 ) )  ->  ( Y  e.  S  /\  X  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  Y ) }
 ) )
 
Theoremgsumbagdiag 16138* Two-dimensional commutation of a group sum over a "triangular" region. fsum0diag 12256 analogue for finite bags. (Contributed by Mario Carneiro, 5-Jan-2015.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  S  =  {
 y  e.  D  |  y  o R  <_  F }   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F  e.  D )   &    |-  B  =  ( Base `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  (
 ( ph  /\  ( j  e.  S  /\  k  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  j ) } )
 )  ->  X  e.  B )   =>    |-  ( ph  ->  ( G  gsumg  ( j  e.  S ,  k  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  j ) }  |->  X ) )  =  ( G  gsumg  ( k  e.  S ,  j  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  k ) }  |->  X ) ) )
 
Theorempsrass1lem 16139* A group sum commutation used by psrass1 16166. (Contributed by Mario Carneiro, 5-Jan-2015.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  S  =  {
 y  e.  D  |  y  o R  <_  F }   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F  e.  D )   &    |-  B  =  ( Base `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  (
 ( ph  /\  ( j  e.  S  /\  k  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  j ) } )
 )  ->  X  e.  B )   &    |-  ( k  =  ( n  o F  -  j )  ->  X  =  Y )   =>    |-  ( ph  ->  ( G  gsumg  ( n  e.  S  |->  ( G  gsumg  ( j  e.  { x  e.  D  |  x  o R  <_  n }  |->  Y ) ) ) )  =  ( G  gsumg  ( j  e.  S  |->  ( G  gsumg  ( k  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  j
 ) }  |->  X ) ) ) ) )
 
Theorempsrbas 16140* The base set of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  K  =  (
 Base `  R )   &    |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  I  e.  V )   =>    |-  ( ph  ->  B  =  ( K  ^m  D ) )
 
Theorempsrelbas 16141* An element of the set of power series is a function on the coefficients. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  K  =  (
 Base `  R )   &    |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  X : D --> K )
 
Theorempsrplusg 16142 The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .+  =  ( +g  `  R )   &    |-  .+b  =  ( +g  `  S )   =>    |-  .+b  =  (  o F  .+  |`  ( B  X.  B ) )
 
Theorempsradd 16143 The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .+  =  ( +g  `  R )   &    |-  .+b  =  ( +g  `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .+b  Y )  =  ( X  o F  .+  Y ) )
 
Theorempsraddcl 16144 Closure of the power series addition operation. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .+  =  ( +g  `  S )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .+  Y )  e.  B )
 
Theorempsrmulr 16145* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  .xb 
 =  ( .r `  S )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   =>    |-  .xb  =  ( f  e.  B ,  g  e.  B  |->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  { y  e.  D  |  y  o R  <_  k }  |->  ( ( f `  x )  .x.  ( g `
  ( k  o F  -  x ) ) ) ) ) ) )
 
Theorempsrmulfval 16146* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  .xb 
 =  ( .r `  S )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   =>    |-  ( ph  ->  ( F  .xb  G )  =  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
 y  e.  D  |  y  o R  <_  k }  |->  ( ( F `
  x )  .x.  ( G `  ( k  o F  -  x ) ) ) ) ) ) )
 
Theorempsrmulval 16147* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  .xb 
 =  ( .r `  S )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |-  ( ph  ->  X  e.  D )   =>    |-  ( ph  ->  (
 ( F  .xb  G ) `
  X )  =  ( R  gsumg  ( k  e.  {
 y  e.  D  |  y  o R  <_  X }  |->  ( ( F `
  k )  .x.  ( G `  ( X  o F  -  k
 ) ) ) ) ) )
 
Theorempsrmulcllem 16148* Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .x.  =  ( .r `  S )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( ph  ->  ( X  .x.  Y )  e.  B )
 
Theorempsrmulcl 16149 Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .x.  =  ( .r `  S )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .x.  Y )  e.  B )
 
Theorempsrsca 16150 The scalar field of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  W )   =>    |-  ( ph  ->  R  =  (Scalar `  S )
 )
 
Theorempsrvscafval 16151* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  .xb  =  ( .s `  S )   &    |-  K  =  ( Base `  R )   &    |-  B  =  ( Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   =>    |-  .xb  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x }
 )  o F  .x.  f ) )
 
Theorempsrvsca 16152* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  .xb  =  ( .s `  S )   &    |-  K  =  ( Base `  R )   &    |-  B  =  ( Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  F  e.  B )   =>    |-  ( ph  ->  ( X  .xb  F )  =  ( ( D  X.  { X } )  o F  .x.  F )
 )
 
Theorempsrvscaval 16153* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  .xb  =  ( .s `  S )   &    |-  K  =  ( Base `  R )   &    |-  B  =  ( Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  Y  e.  D )   =>    |-  ( ph  ->  (
 ( X  .xb  F ) `
  Y )  =  ( X  .x.  ( F `  Y ) ) )
 
Theorempsrvscacl 16154 Closure of the power series scalar multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  .x.  =  ( .s `  S )   &    |-  K  =  ( Base `  R )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  F  e.  B )   =>    |-  ( ph  ->  ( X  .x.  F )  e.  B )
 
Theorempsr0cl 16155* The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  B  =  ( Base `  S )   =>    |-  ( ph  ->  ( D  X.  {  .0.  }
 )  e.  B )
 
Theorempsr0lid 16156* The zero element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  B  =  ( Base `  S )   &    |-  .+  =  ( +g  `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  (
 ( D  X.  {  .0.  } )  .+  X )  =  X )
 
Theorempsrnegcl 16157* The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  N  =  ( inv g `  R )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( N  o.  X )  e.  B )
 
Theorempsrlinv 16158* The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  N  =  ( inv g `  R )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  .0.  =  ( 0g `  R )   &    |- 
 .+  =  ( +g  `  S )   =>    |-  ( ph  ->  (
 ( N  o.  X )  .+  X )  =  ( D  X.  {  .0.  } ) )
 
Theorempsrgrp 16159 The ring of power series is a group. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   =>    |-  ( ph  ->  S  e.  Grp )
 
Theorempsr0 16160* The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  O  =  ( 0g `  R )   &    |-  .0.  =  ( 0g `  S )   =>    |-  ( ph  ->  .0.  =  ( D  X.  { O } ) )
 
Theorempsrneg 16161* The negative function of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  N  =  ( inv g `  R )   &    |-  B  =  ( Base `  S )   &    |-  M  =  ( inv g `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( M `  X )  =  ( N  o.  X ) )
 
Theorempsrlmod 16162 The ring of power series is a left module. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  S  e.  LMod
 )
 
Theorempsr1cl 16163* The identity element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  U  =  ( x  e.  D  |->  if ( x  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) )   &    |-  B  =  ( Base `  S )   =>    |-  ( ph  ->  U  e.  B )
 
Theorempsrlidm 16164* The identity element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  U  =  ( x  e.  D  |->  if ( x  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) )   &    |-  B  =  ( Base `  S )   &    |-  .x.  =  ( .r `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( U  .x.  X )  =  X )
 
Theorempsrridm 16165* The identity element of the ring of power series is a right identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  U  =  ( x  e.  D  |->  if ( x  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) )   &    |-  B  =  ( Base `  S )   &    |-  .x.  =  ( .r `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( X  .x.  U )  =  X )
 
Theorempsrass1 16166* Associative identity for the ring of power series. (Contributed by Mario Carneiro, 5-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .X.  =  ( .r `  S )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  ( ( X  .X.  Y )  .X.  Z )  =  ( X  .X.  ( Y  .X.  Z ) ) )
 
Theorempsrdi 16167* Distributive law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .X.  =  ( .r `  S )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  .+  =  ( +g  `  S )   =>    |-  ( ph  ->  ( X  .X.  ( Y  .+  Z ) )  =  ( ( X  .X.  Y )  .+  ( X  .X.  Z ) ) )
 
Theorempsrdir 16168* Distributive law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .X.  =  ( .r `  S )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  .+  =  ( +g  `  S )   =>    |-  ( ph  ->  ( ( X 
 .+  Y )  .X.  Z )  =  ( ( X  .X.  Z )  .+  ( Y  .X.  Z ) ) )
 
Theorempsrcom 16169* Commutative law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .X.  =  ( .r `  S )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  R  e.  CRing )   =>    |-  ( ph  ->  ( X  .X.  Y )  =  ( Y  .X.  X ) )
 
Theorempsrass23 16170* Associative identities for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .X.  =  ( .r `  S )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  K  =  (
 Base `  R )   &    |-  .x.  =  ( .s `  S )   &    |-  ( ph  ->  A  e.  K )   =>    |-  ( ph  ->  (
 ( ( A  .x.  X )  .X.  Y )  =  ( A  .x.  ( X  .X.  Y ) ) 
 /\  ( X  .X.  ( A  .x.  Y ) )  =  ( A 
 .x.  ( X  .X.  Y ) ) ) )
 
Theorempsrrng 16171 The ring of power series is a ring. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  S  e.  Ring
 )
 
Theorempsr1 16172* The identity element of the ring of power series. (Contributed by Mario Carneiro, 8-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  U  =  ( 1r `  S )   =>    |-  ( ph  ->  U  =  ( x  e.  D  |->  if ( x  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) ) )
 
Theorempsrcrng 16173 The ring of power series is commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  CRing )   =>    |-  ( ph  ->  S  e.  CRing
 )
 
Theorempsrassa 16174 The ring of power series is an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  CRing )   =>    |-  ( ph  ->  S  e. AssAlg )
 
Theoremresspsrbas 16175 A restricted power series algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPwSer  H )   &    |-  B  =  (
 Base `  U )   &    |-  P  =  ( Ss  B )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   =>    |-  ( ph  ->  B  =  ( Base `  P )
 )
 
Theoremresspsradd 16176 A restricted power series algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPwSer  H )   &    |-  B  =  (
 Base `  U )   &    |-  P  =  ( Ss  B )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X ( +g  `  U ) Y )  =  ( X ( +g  `  P ) Y ) )
 
Theoremresspsrmul 16177 A restricted power series algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPwSer  H )   &    |-  B  =  (
 Base `  U )   &    |-  P  =  ( Ss  B )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X ( .r `  U ) Y )  =  ( X ( .r `  P ) Y ) )
 
Theoremresspsrvsca 16178 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPwSer  H )   &    |-  B  =  (
 Base `  U )   &    |-  P  =  ( Ss  B )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   =>    |-  ( ( ph  /\  ( X  e.  T  /\  Y  e.  B )
 )  ->  ( X ( .s `  U ) Y )  =  ( X ( .s `  P ) Y ) )
 
Theoremsubrgpsr 16179 A subring of the base ring induces a subring of power series. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPwSer  H )   &    |-  B  =  (
 Base `  U )   =>    |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R )
 )  ->  B  e.  (SubRing `  S ) )
 
Theoremmvridlem 16180* A bag containing one element is a finite bag. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   =>    |-  ( I  e.  V  ->  ( y  e.  I  |->  if ( y  =  X ,  1 ,  0 ) )  e.  D )
 
Theoremmvrfval 16181* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  V  =  ( I mVar 
 R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Y )   =>    |-  ( ph  ->  V  =  ( x  e.  I  |->  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) ) )
 
Theoremmvrval 16182* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  V  =  ( I mVar 
 R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Y )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  ( V `  X )  =  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  X ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) )
 
Theoremmvrval2 16183* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  V  =  ( I mVar 
 R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Y )   &    |-  ( ph  ->  X  e.  I )   &    |-  ( ph  ->  F  e.  D )   =>    |-  ( ph  ->  ( ( V `  X ) `  F )  =  if ( F  =  ( y  e.  I  |->  if ( y  =  X ,  1 ,  0 ) ) ,  .1.  ,  .0.  )
 )
 
Theoremmvrid 16184* The  X i-th coefficient of the term  X i is  1. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  V  =  ( I mVar 
 R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Y )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  (
 ( V `  X ) `  ( y  e.  I  |->  if ( y  =  X ,  1 ,  0 ) ) )  =  .1.  )
 
Theoremmvrf 16185 The power series variable function is a function from the index set to elements of the power series structure representing  X
i for each  i. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  V  =  ( I mVar  R )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  V : I --> B )
 
Theoremmvrf1 16186 The power series variable function is injective if the base ring is nonzero. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  V  =  ( I mVar  R )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  .1. 
 =/=  .0.  )   =>    |-  ( ph  ->  V : I -1-1-> B )
 
Theoremmvrcl2 16187 A power series variable is an element of the base set. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  V  =  ( I mVar  R )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  ( V `  X )  e.  B )
 
Theoremreldmmpl 16188 The multivariate polynomial constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- 
 Rel  dom mPoly
 
Theoremmplval 16189* Value of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  R )   &    |-  U  =  { f  e.  B  |  ( `' f " ( _V  \  {  .0.  } )
 )  e.  Fin }   =>    |-  P  =  ( Ss  U )
 
Theoremmplbas 16190* Base set of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  R )   &    |-  U  =  ( Base `  P )   =>    |-  U  =  { f  e.  B  |  ( `' f " ( _V  \  {  .0.  } )
 )  e.  Fin }
 
Theoremmplelbas 16191 Property of being a polynomial. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  R )   &    |-  U  =  ( Base `  P )   =>    |-  ( X  e.  U  <->  ( X  e.  B  /\  ( `' X " ( _V  \  {  .0.  } )
 )  e.  Fin )
 )
 
Theoremmplval2 16192 Self-referential expression for the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  U  =  (
 Base `  P )   =>    |-  P  =  ( Ss  U )
 
Theoremmplbasss 16193 The set of polynomials is a subset of the set of power series. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  U  =  (
 Base `  P )   &    |-  B  =  ( Base `  S )   =>    |-  U  C_  B
 
Theoremmplelf 16194* A polynomial is defined as a function on the coefficients. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  K  =  (
 Base `  R )   &    |-  B  =  ( Base `  P )   &    |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  X : D --> K )
 
Theoremmplsubglem 16195* If  A is an ideal of sets (a nonempty collection closed under subset and binary union) of the set  D of finite bags (the primary applications being  A  =  Fin and  A  =  ~P B for some  B), then the set of all power series whose coefficient functions are supported on an element of  A is a subgroup of the set of all power series. (Contributed by Mario Carneiro, 12-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  R )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  (/)  e.  A )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A )
 )  ->  ( x  u.  y )  e.  A )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  C_  x ) ) 
 ->  y  e.  A )   &    |-  ( ph  ->  U  =  { g  e.  B  |  ( `' g "
 ( _V  \  {  .0.  } ) )  e.  A } )   &    |-  ( ph  ->  R  e.  Grp )   =>    |-  ( ph  ->  U  e.  (SubGrp `  S )
 )
 
Theoremmpllsslem 16196* If  A is an ideal of subsets (a nonempty collection closed under subset and binary union) of the set  D of finite bags (the primary applications being  A  =  Fin and  A  =  ~P B for some  B), then the set of all power series whose coefficient functions are supported on an element of  A is a linear subspace of the set of all power series. (Contributed by Mario Carneiro, 12-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  R )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  (/)  e.  A )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A )
 )  ->  ( x  u.  y )  e.  A )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  C_  x ) ) 
 ->  y  e.  A )   &    |-  ( ph  ->  U  =  { g  e.  B  |  ( `' g "
 ( _V  \  {  .0.  } ) )  e.  A } )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  U  e.  ( LSubSp `  S )
 )
 
Theoremmplsubg 16197 The set of polynomials is closed under addition, i.e. it is a subgroup of the set of power series. (Contributed by Mario Carneiro, 8-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  P  =  ( I mPoly  R )   &    |-  U  =  ( Base `  P )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Grp )   =>    |-  ( ph  ->  U  e.  (SubGrp `  S )
 )
 
Theoremmpllss 16198 The set of polynomials is closed under scalar multiplication, i.e. it is a linear subspace of the set of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  P  =  ( I mPoly  R )   &    |-  U  =  ( Base `  P )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  U  e.  ( LSubSp `  S )
 )
 
Theoremmplsubrglem 16199* Lemma for mplsubrg 16200. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  P  =  ( I mPoly  R )   &    |-  U  =  ( Base `  P )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  {
 f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  A  =  (  o F  +  " ( ( `' X " ( _V  \  {  .0.  } )
 )  X.  ( `' Y " ( _V  \  {  .0.  } ) ) ) )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   =>    |-  ( ph  ->  ( X ( .r `  S ) Y )  e.  U )
 
Theoremmplsubrg 16200 The set of polynomials is closed under multiplication, i.e. it is a subring of the set of power series. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  P  =  ( I mPoly  R )   &    |-  U  =  ( Base `  P )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  U  e.  (SubRing `  S )
 )
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